I have iMessage on my Mac and iPhone.
I treat iMessage on my phone as the primary tool. Mac iMessage is convenient to send messages occasionally, but my Mac is not always connected to the internet.
A message has been delivered to my Mac, and after several hours it has not appeared on my phone. I have tried logging out of iMessage. Turning iMessage off. restarting by holding down home and power. waiting ten minutes and reconnecting.
Can't make it budge. I have had an ongoing conversation since then.
Best Answer
I have no solid reference to back this up, but I have seen it empirically many times over many years & devices.
Once a message has been received & read by your account, no matter what device, other devices will no longer receive that message.
There may be a timeout period - as you can clearly see when you have your phone & Mac next to each other, receiving at the same time. The one that last 'spoke' will be the one on which an incoming message will actually 'ping' & the other will wait about a minute before also pinging...
...But if one is off [& this is the bit about which I have no real information ... how long it needs to have been off & remain off], & the message is received & read on the other, then the first will never receive the message at all.
The same can be seen if your phone is outside reception range & a message is received on the Mac [at home]. You only discover the message when you get back home.
See related: How to tell Messages "I'm not at home"?